Treatment protocol generation for diseases related to angiogenesis

ABSTRACT

A computer-implemented method for determining an optimal treatment protocol for a disease related to angiogenesis, comprising creating an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors. Effective vessel density (EVD) is incorporated as a factor regulating switching on and switching off of at least one component in the angiogenesis model. Effects of vasculature maturation and mature vessels destabilization are incorporated. Pro-angiogenesis and anti-angiogenesis factors, which can influence changes in state of a tissue are selected. Effects of drugs in the pro-angiogenesis and anti-angiogenesis factors are incorporated. A plurality of treatment protocols in a protocol space is generated. A best treatment protocol based on a pre-determined criteria.

RELATED APPLICATIONS

This Application is a continuation of co-pending U.S. application Ser.No. 10/207,772, filed on Jul. 31, 2002, which in turn claims benefit ofco-pending U.S. Provisional Patent Application Ser. No. 60/330,592 filedOct. 25, 2001, the contents of both of which Applications areincorporated herein by reference.

FIELD

The present disclosure generally teaches techniques related to diseasesand processes involving Angiogenesis. More particularly it teachestechniques for generating treatment protocols for diseases whereangiogenesis is a factor. The techniques are also applicable to normalprocesses involving Angiogenesis even if no disease is present.

BACKGROUND

-   -   1. References

The following papers provide useful background information, for whichthey are incorporated herein by reference in their entirety, and areselectively referred to in the remainder of this disclosure by theiraccompanying reference numbers in brackets (i.e., <3> for the thirdnumbered paper by Yangopoulos et al):

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<28> Lindahl P, Johansson B R, Leveen P, Betsholtz C (1997) PericyteLoss and Microaneurysm Formation in PDGF-B- Deficient Mice. Science 277:242-245 11 Jul. 1997

<29> Hellstrom M, Kal'n M, Lindahl P, Abramsson A, Betsholtz C (1999)Role of PDGF-B and PDGFR-b in recruitment of vascular smooth musclecells. Development 126: 3047-3055

<30> Davis S, Aldrich T H, Jones P F, Acheson A, Compton D L, Jain V,Ryan T E, Bruno J, Radziejewski C, Maisonpierre P C, Yancopoulos G D(1996) Isolation of Angiopoietin-1, a Ligand for the TIE2 Receptor, bySecretion-Trap Expression Cloning. Cell 87: 1161-1169. December 1996

<31> Suri C, Jones P E, Patan S, Bartunkova S, Maisonpierre P C, DavisS, Sato T S, Yancopoulos G D (1996) Requisite Role of Angiopoietin-1, aLigand for the TIE2 Receptor, during Embryonic Angiogenesis. Cell 87:1171-1180. December 1996

<32> Maisonpierre P C, Suri C, Jones P F, Bartunkova S, Wiegand S J,Radziejewski C, Compton D, McClain J, Aldrich T H, Papadopoulos N, DalyT J, Davis S, Sato T N, Yancopoulos G D (1997) Angiopoietin-2, a NaturalAntagonist for Tie2 That Disrupts in vivo Angiogenesis. Science 277.5322: 55-60. 4 Jul 1997

<33> Folkman J, D'Amore P A (1996) Blood Vessel Formation: What Is ItsMolecular Basis?. Cell 87: 1153-1155. December 1996

<34> Eggert A, Ikegaki N, Kwiatkowski J, Zhao H, Brodeur G M, HimelsteinB P (2000) High-Level Expression of Angiogenic Factors Is Associatedwith Advanced Tumor Stage in Human Neuroblastomas. Clinical CancerResearch 6: 1900-1908

<35> Loughna S. and Sato T N. (2001) Angiopoietin and Tie signalingpathways in vascular development. Matrix Biol. 20:319-25.

<36> Folkman, J. (1971) Tumor angiogenesis: therapeutic implications. N.Engl. J. Med. 285:1182-1186.

<37> O'Reilly, M. S. et al.(1994) Angiostatin: a novel angiogenesisinhibitor that mediates the suppression of metastases by a Lewis lungcarcinoma. Cell. 79:315-328.

<38> O'Reilly, M. S. et al.(1997) Endostatin: an endogenous inhibitor ofangiogenesis and tumor growth. Cell. 88:277-285.

<39> Anderson A R and Chaplain M A (1998) Continuous and discretemathematical models of tumor-induced angiogenesis. Bull Math Biol.60(5):857-99.

<40> Hahnfeldt P. et al. (1999) Tumor development under angiogenicsignaling: a dynamic theory of tumor growth, treatment response, andpostvascular dormancy. Cancer Res. 59:4770-5.

<41> Rolland, Y. et al. (1998) Modeling of the parenchymousvascularization and perfusion. Invest. Radiology 34:171-5.

<42> Hayes, A J et al. (2000) Expression and function of angiopoietin-1in breast cancer. Br. J. Cancer 83:1154-60.

<43> Ahmad, S A et al. (2001) Differential expression of angiopoietin-1and angiopoietin-2 in colon carcinoma. Cancer 92:1138-43.

<44> Koga K. et al. (2001) Expression of angiopoietin-2 in human gliomacells and its role for angiogenesis. Cancer Res. 61:6248-54.

<45> Oh H. et al. (1999) Hypoxia and Vascular Endothelial Growth Factorselectively up-regulate angiopoietin-2 in bovine microvascularendothelial cells. J Biol Chem 274:15732-9.

<46> Holash J. et al. (1999) Vessel cooption, regression, and growth intumors mediated by angiopoietins and VEGF. Science 284:1994-8.

<47> Mandriota S J et al. (2000) Hypoxia-inducible angiopoietin-2expression is mimicked by lodonium compounds and occurs in the rat brainand skin in response to systemic hypoxia and tissue ischemia. Am JPathol 156:2077-89.

<48> Beck H. (2000) Expression of angiopoietin-1, angiopoietin-2, andTie receptors after middle cerebral artery occlusion in rats. Am JPathol 157:1473-83.

<49> Stratmann A. et al. (1998) Cell type-specific expression ofangiopoietin-1 and angiopoietin-2 suggests a role in glioblastomaangiogenesis. Am J Pathol 153:1459-66.

<50> Stegmann T J et al. (2000) [Induction of myocardial neoangiogenesisby human growth factors. A new therapeutic approach in coronary heartdisease] Herz.; 25(6):589-99.

<51> Kocher A A et al. (2001) Neovascularization of ischemic myocardiumby human bone-marrow-derived angioblasts prevents cardiomyocyteapoptosis, reduces remodeling and improves cardiac function. NatMed.;7(4):430-6.

<52> Ciulla T A et al. (2001) Presumed ocular histoplasmosis syndrome:update on epidemiology, pathogenesis, and photodynamic, antiangiogenic,and surgical therapies. Curr Opin Ophthalmol.; 12(6):442-9.

<53> Weber A J, De Bandt M. (2000) Angiogenesis: general mechanisms andimplications for rheumatoid arthritis. Joint Bone Spine.; 67(5):366-83.

<54> Fearon U, Veale D J. (2001) Pathogenesis of psoriatic arthritis.Clin Exp Dermatol.; 26(4): 333-7.

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<56> Gustafsson T., Kraus W E. (2001) Exercise-inducedangiogenesis-related growth and transcription factors in skeletalmuscle, and their modification in muscle pathology. Front Biosci; 6:D75-89.

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-   -   2. Introduction

For a better understanding of this disclosure, all the terms andparameters used in this disclosure are listed in the Table shown in FIG.5.

Angiogenesis, or neovascularization, is a process of new blood vesselsformation by budding from the existing ones. Neovascularization providestissue with vital nutrients and growth factors and enables clearance oftoxic waste products of cellular metabolism. Angiogenesis has beenconventionally recognized as a biological mechanism of dual clinicaleffect. On the one hand, it allows survival of normal tissues when theybecome ischemic. That is, it enables functional development of normaltissues, for example, wound healing and embryogenesis. On the otherhand, angiogenesis enables tumor tissue growth.

Intensive research of angiogenesis during the last 15 years has led tobetter understanding of this complex process <1-7, 57-60>. However,cause and effect relationships in the process of angiogenesis are yet tobe clarified. Moreover, the massive research in the field of angiogenictherapy still suffers from the lack of tools for predicting thepotential effects of PRO-and ANTI-angiogenic factors.

The two major determinants of new vasculature formation are thought tobe the genetic features of the tissue and the availability of oxygen andnutrients <5,6>. The dependence of vessel formation on nutrients oroxygen deprivation was shown to be mediated by vascular endothelialgrowth factor (VEGF), which is a potent inducer of endothelial cellproliferation and migration <8-15>. VEGF is preferentially expressed bytissue cells in the nutrient-deprived areas <7, 16-20, 61-64>. Incontrast, basic nutrient-independent VEGF production by the tissue isdetermined by genetic factors <20>. Consequently, VEGF-inducedangiogensis depends on both the aforementioned vasculature growthdeterminants, namely genetics and nutrient/oxygen availability.VEGF-induced angiogenesis leads to increase in nutrient supply to thetissue. Accordingly, nutrient- and oxygen-dependent VEGF expression isdown-regulated. When VEGF level becomes low enough, the newly formedblood vessels regress <21-26>, consequently leading to nutrients andoxygen deprivation again. This negative feedback can produce successivecycles of growth and regression of blood vessels. This phenomenon wasdemonstrated in the mouse xenograft tumor model <12>.

Blood vessels can be rendered insensitive to fluctuations of VEGFconcentration by the process of maturation (coverage of capillaries byperiendothelial cells 25,26). This process involves pericytes (smoothmuscle-like cells) which form an outside layer covering the endothelialcells of the newly formed vessel . The major pericyte-stimulating factoris a platelet-derived growth factor (PDGF) <27-29>. Interactions betweenendothelial cells and pericytes, which apparently lead to maturation,are governed by the angiopoietin system. This system includes twosoluble factors—Angiopoietins 1 and 2 (Ang1 and Ang2, respectively), andtheir receptor, Tie2, which is specifically expressed on endothelialcells <30-34>. Ang1 is Tie2 agonist that promotes maturation, while Ang2is its natural antagonist <31,32>. Regulation of Ang1 and Ang2expression is not completely understood. According to recentpublications it can be influenced by tumor cell- as well as endothelialcell-specific factors <35,42-49>. These factors depend on the tissue andthe host type <42-44>, and can be, accordingly, taken into account inthe presented model. High Ang 1/Ang 2 ratio and pericytes' presenceinduces maturation of newly formed vessels. Alternatively, low Ang 1/Ang2 ratio induces destabilization of mature blood vessels, while newlyformed vessels remain immature and susceptible to VEGF fluctuations<25>.

Therefore, it would be advantageous to have techniques for generatingand selecting treatment protocols for diseases where angiogenesis is afactor. Further, it is also advantageous to adapt the techniques tostudy the progression of processes involving angiogenesis.

-   -   3. Clinical Significance

The clinical significance of angiogenesis as an “ultimate” target forcancer therapy was first recognized in 1971 by J. Folkman <36>, and gotwide acceptance in early nineties after the discovery of the firstspecific antiangiogenic substances <37,38>. Apparent advantages of thisapproach include its universality for different solid tumors, lack ofprominent side effects and lack of resistance development duringrepetitive treatment cycles.

Angiogenesis is implicated in the pathogenesis of a variety ofdisorders: proliferative retinopathies, age-related maculardegeneration, tumors, rheumatoid arthritis, psoriasis <1; 51-56, 66, 67>and coronary heart diseases <50>. The use of exogenous agents toselectively target neovasculature, or stimulate the growth of new bloodvessels into ischaemic tissue is a potentially revolutionary therapy ina wide variety of clinical specialties, which opens new avenues for thediagnosis and therapy of diseases where angiogenesis is a factor; suchas, cancer, blinding ocular disorders, rheumatoid arthritis andothers.<65, 66>.

-   -   4. Problems in Practicing Anti/Pro-Angiogenic Therapy

In order to establish optimal pro-angiogenic or anti-angiogenictreatment protocols (either as a monotherapy or in combination withchemotherapy or radiotherapy) the dynamics of angiogenesis must bebetter understood. Recent studies <25,26> have shown that newly formedvasculature is very dynamic—blood vessels undergo constant remodelingthat involves maturation in response to local levels of angiogenic andmaturation factors. Mature and immature vessels may differentiallyrespond to certain PRO and ANTI-angiogenic drugs during tissue growth,myocardial ischemia, macular degeneration and other diseases, leading tosuccess or failure of the treatment <25>.

Mathematical models and computer simulation of angiogenesis and PRO- andANTI-angiogenic therapy can be constructed, in order to predict the mostpromising treatment protocols thus eliminating the need for lengthy andexpensive clinical trials.

-   -   5. Previous Mathematical Models

The construction of a mathematical model for angiogenesis includes I)in-depth understanding of the biology of angiogenesis, II) the selectionof appropriate patient populations for clinical trials, choice oftherapeutic end points and means of their assessment, choice oftherapeutic strategy (gene versus protein delivery), route ofadministration, and the side effect profile.<68>

Several mathematical dynamic models have been proposed, each one of themconstructed to illuminate specific aspects of angiogenesis <39-41>. Someof these models examine vascular tree formation in vitro, irrespectivelyof tumor dynamics, and consequently are not suitable for tumor growthmodelling <39>. Others assume that the growing vascular tree is asubject to some optimization with regard to the target tissue perfusion<41>. This optimization, while possibly holding true for normal tissuedevelopment, can hardly account for tumor growth, since it is known thattumor vasculature is highly disorganized.

Logistic-type model, proposes by Hahnfeldt et al. <40>, analyzes thegeneral vascular dynamics (“carrying capacity of current vascular tree”)with regard to production of pro- and antiangiogenic factors by thetumor. Analysis of experimental data of Lewis lung carcinoma growth inmice allowed the authors to estimate the model parameters and to examinethe effects of antiangiogenic factors angiostatin and endostatin. Themain problematic assumption of this model is the constant productionrates of these factors, as we know that VEGF, for example, production istightly regulated by tissue hypoxia.

Model by Tong S and Yuan F <69> focused on two-dimensional angiogenesisin the cornea. The model considered diffusion of angiogenic factors,uptake of these factors by endothelial cells, and randomness in the rateof sprout formation and the direction of sprout growth.

None of the aforementioned models takes into account vasculaturematuration and mature vessels destabilization, which are veryfundamental constituents of angiogenesis dynamics. Moreover, thesemodels, due to their relative abstraction, cannot account fordrug-induced, or other, molecular changes in angiogenic dynamics. Note,that, since PRO and ANTI-angiogenic drugs interfere with the dynamicsdescribed above at the molecular level, the model which can serve as atool for predicting drug effect on this process must take into accountall the molecular complexity of angiogenesis, including the dynamics ofneovasculature maturation and mature vessels destabilisation.

Therefore, it is desirable to provide a techniques, including computersystems, that overcomes some of the disadvantages noted above.

SUMMARY

To realize the advantages discussed above, the disclosed teachingsprovide a computer-implemented method for determining an optimaltreatment protocol for a disease related to angiogenesis, comprisingcreating an angiogenesis model including pro-angiogenesis andanti-angiogenesis factors. Effective vessel density (EVD) isincorporated as a factor regulating switching on and switching off of atleast one component in the angiogenesis model. Effects of vasculaturematuration and mature vessels destabilization are incorporated.Pro-angiogenesis and anti-angiogenesis factors, which can influencechanges in state of a tissue are selected. Effects of drugs in thepro-angiogenesis and anti-angiogenesis factors are incorporated. Aplurality of treatment protocols in a protocol space is generated. Abest treatment protocol based on a pre-determined criteria.

In another specific enhancement, the model comprises a tissue volumemodel, an immature vessel model and a mature vessel model.

In another specific enhancement, steps to regulate dynamics whichinfluences EVD are incorporated.

In another specific enhancement, the model simultaneously accounts fortissue cell proliferation, tissue cell death, endothelial cellproliferation, endothelial cell death, immature vessel formation andimmature vessel regression, immature vessel maturation and mature vesseldestabilization.

In another specific enhancement, the model incorporates temporalparameters that characterize response rate of at least one elementassociated with angiogenesis.

More specifically, EVD is calculated by combining immature vesseldensity and mature vessel density.

In another specific enhancement, parameters incorporated into the modelcomprises tissue volume, number of free endothelial cells, number offree pericytes, volume of mature vessels, volume of immature vessels andconcentration of regulator factors.

More specifically, the regulatory factors comprise vascular endothelialgrowth factor (VEGF), platelet derived growth factor (PDGF),angiopoietin 1 (Ang1) and angiopoietin 2 (Ang2).

More specifically, EVD is a function of a duration of insufficientperfusion and vice versa .

More specifically, the model incorporates threshold levels of regulatoryfactors and parameter ratios.

Even more specifically, the threshold level is at least one of: a) VEGFconcentration below which no endothelial cells proliferation takes place(A); b) minimum number of receptors for VEGF above which endothelialcells proliferation takes place (B); c) VEGF concentration below whichendothelial cells, both in the free state as well as when incorporatedinto immature blood vessels, are subject to apoptosis VEGF_(thr); d) theminimal number of free pericytes which stimulates the onset ofmaturation of immature vessels (C); e) Ang 1/Ang 2 ratio below whichmature vessels are destabilized, and above which maturation of immaturevessels is enabled (K); f) EVD value that influences the rate of cellproliferation and death; and g) EVDS_(ss) value for which the system isin steady state. Even more specifically, the tissue volume modelcalculates the tissue volume by a process comprising: comparing EVDagainst an −EVD_(ss). If EVD is equal to EVD_(ss) then use a programmedtissue cell proliferation and a programmed tissue cell death (apoptosis)to compute tissue volume. If EVD >EVD_(ss) then use increased tissueproliferation and decreased tissue cell death to compute tissue volume.If EVD<EVD_(ss) then use decreased tissue proliferation and increasedtissue cell death to compute tissue volume.

Even more specifically, Ang1 and Ang2 induction are incorporated intoappropriate steps above following the computation of tissue volume.

More specifically, the immature vessel model calculates the immaturevessels by a process comprising comparing EVD against an EVD_(ss). IfEVD is equal to EVD_(ss) then set VEGF to a VEGF_(ss) and PDGF to aPDGF_(ss). If EVD>EVD_(ss) then use decreased VEGF and decreased PDGF.If EVD<EVD_(ss) then useg increased VEGF and increased PDGF. CompareVEGF against A. Factor endothelial cell proliferation if VEGF>A. CompareVEGF against a VEGF threshold. Factor free endothelial cell deaths ifVEGF<VEGF threshold. Compare VEGF receptor number against B. If VEGFreceptor number is less than B then consider no angiogenisis prior tocomputing immature vessel regression. If VEGF receptor number is notless than B then compute growth of immature vessels. If VEGF<A thenconsider no angiogenesis and compute immature vessel regression. Computemature vessels based on growth immature vessels, immature vesselregression and mature vessel destabilization.

Even more specifically, immature vessels computation considers nomaturation if Ang2/Ang1>K or if number of free pericytes<C.

Even more specifically, mature vessel destabilization considersang1/Tie2 interaction blocking.

Even more specifically, no destabilization occurs if Ang2/Ang1 is notgreater than K

More specifically, mature vessel model is computed using a procedurecomprising computing immature vessels. Determining if Ang1/Ang2<K.Determine if number of free pericytes<C. Considering immature vesselmaturation if both the above steps are false. Factoring nodestablization if number of free pericytes is not less than C.

More specifically, effects of a drug affecting EC proliferation arefactored in computing immature vessels.

More specifically, effects of a drug affecting VEGF receptors arefactored in computing immature vessels.

More specifically, effects of a drug affecting pericyte proliferationare factored in computing immature vessel computation.

More specifically, effects of a drug affecting VEGF are factored incomputing immature vessels.

More specifically, effects of a drug affecting PEGF are factored incomputing immature vessel computation.

More specifically, effects of a drug affecting Ang1 are factored incomputing immature vessels.

More specifically, effects of a drug affecting Ang2 are factored incomputing immature vessel computation.

In another specific enhancement, model takes into account duration of atissue cell proliferation, tissue cell death, endothelial cellproliferation, endothelial cell death, pericytes proliferation, immaturevessel regression, immature vessel maturation and mature vesseldestabilization.

In another specific enhancement, model takes into account the durationof VEGF induction, PDGF induction, Ang1 and Ang2 induction by tissuecells and Ang1 and Ang2 induction by endothelial cells.

Another aspect of the disclosed teachings is an optimal treatmentprotocol for a disease related to angiogenesis, comprising anangiogenesis model including pro-angiogenesis and anti-angiogenesisfactors; a treatment protocol space generator that generates a protocolspace of possible treatments for the disease; a treatment selector thatselects an optimal protocol, wherein effective vessel density (EVD) is afactor regulating switching on and switching off of at least onecomponent in the angiogenesis model; wherein the model incorporateseffects of vasculature maturation and mature vessels destabilization;and wherein the system is adapted to effect selection ofpro-angiogenesis and anti-angiogenesis factors which can influencechanges in state of a tissue and incorporating effects of drugs in thepro-angiogenesis and anti-angiogenesis factors.

A computer program product including computer readable media thatcomprises instructions to implement the above techniques on a computerare also part of the disclosed teachings.

BRIEF DESCRIPTION OF THE DRAWINGS

The above objectives and advantages of the disclosed teachings willbecome more apparent by describing in detail preferred embodimentthereof with reference to the attached drawings in which:

FIG. 1 (a)-(c) depicts a flowchart that shows an implementation of theoverall techniques.

FIG. 2 (a)-(c) depicts the flowchart of FIG. 1 with possible effects dueto Pro-Angiogeneses and Anti-Angiogeneses drugs.

FIG. 3 (a)-(f) shows graphs that illustrate the effects of Anti-VEGFDrugs.

FIG. 4 shows graphs that illustrate the effects of a combination ofAnti-VEGF and Anti-Ang1 Drugs.

FIG. 5 (a)-(c) shows a table with description of terms used in theEquations included in the specification.

DETAILED DESCRIPTION

IV.A. Overview of Exemplary Implementations

The disclosed techniques are embodied in exemplary computer systems andexemplary flowcharts that are implemented by computers. Theimplementations discussed herein are merely illustrative in nature andare by no means intended to be limiting. Also it should be understoodthat any type of computers can be used to implement the systems andtechniques. An aspect of the disclosed teachings is a computer programproduct including computer-readable media comprising instructions. Theinstructions are capable of enabling a computer to implement the systemsand techniques described herein. It should be noted that thecomputer-readable media could be any media from which a computer canreceive instructions, including but not limited to hard disks, RAMs,ROMs, CDs, magnetic tape, internet downloads, carrier wave with signals,etc. Also instructions can be in any form including source code, objectcode, executable code, and in any language including higher level,assembly and machine languages. The computer system is not limited toany type of computer. It could be implemented in a stand-alone machineor implemented in a distributed fashion, including over the internet.

The technique shown in the flowchart take into account for the dynamicinteractions between tissue volume, angiogenesis (growth and regressionof immature blood vessels), and vascular maturation and destabilization.The technique shown in the flowchart is combined with a quantitativemathematical model that is described in detail herein. A combination ofthe technique shown in the flowchart and the mathematical computationsdescribed would allow a skilled artisan to practice the disclosedtechnique; including for example, to quantify the dynamics of tissuevascularization and the effect of drug on this process at any givenmoment.

The technique describes the interactions between molecular regulatoryfactors, cell types and multi-cellular structures (such as vessels)which together influence the tissue dynamics.

The technique takes into account, the temporal parameters whichcharacterize the response rates of each one of the elements included inthe angiogenesis process.

The technique includes a series of simulation steps. The parametervalues that are output from each simulation step are taken as initialconditions for the next simulation step. These parameter values arecompared with the threshold levels. Their current values are calculatedaccording to the arrows shown in the flowchart of FIG. 1. At least sixmajor processes are taken into account simultaneously, namely tissuecells proliferation and death, endothelial cells proliferation anddeath, immature vessels formation and regression, immature vesselsmaturation and mature vessels destabilization, and possibly others.

The techniques depicted in FIG. 1 includes three interconnected modules:tissue cell proliferation, angiogenesis (immature vessels growth) andmaturation (formation and destabilization of mature vessels). Further,each module operates on three scales: molecular, cellular andmacroscopic (namely, vessel densities and tissue volume).

The tissue module includes tissue cell proliferation sub-module and celldeath sub-module. Further, each is subdivided into i) time-invariant,cell type-specific, genetically determined sub-block, and ii)time-variant, nutrient-dependent sub-block. Nutrient-dependent cellproliferation and nutrient-dependent cell death rates are directly orinversely proportional, respectively, to the effective vascular density(EVD), which is a perfused part of vascular tree <40>.

Two additional quantities are calculated in the tissue module, namelyVEGF and PDGF production. They are inversely related to EVD so thatincreasing nutrient depletion results in increasing secretion ofpro-angiogenic factors <7-9>. The tissue growth module interacts withthe angiogenesis and the maturation modules via the relevant regulatoryproteins.

In the angiogenesis module, volume of immature vessels is calculated.Immature vessel volume increases proportionally to VEGF concentration,if VEGF is above a given threshold level. The volume regresses if VEGFis below a given, possibly different, threshold level. The latterthreshold is generally referred to as “survival level” <21-24>.

In the maturation module, volume of mature vessels is calculatedaccording to pericyte concentration <41-43>and according to theAng1/Ang2 ratio <44>. Pericytes proliferate proportionally to PDGFconcentration <25-26>. Ang1 and Ang2 are continuously secreted by tissuecells and immature vessels, respectively <27, 28, 32-34, 41-43, 45>.Additionally, Ang1 and Ang2 can be secreted by tissue cells, if thelatter are nutrient-depleted <45>. It is assumed that maturation ofimmature vessels occurs if pericytes concentration and Ang1/Ang2 ratioare above their respective threshold levels, while under thesethresholds immature vessels do not undergo maturation, while maturevessels undergo destabilization and become immature <29-33>.

It is clear that the parameters used in the technique can include tissuevolume (determined as a function of tissue cell number); number of freeendothelial cells and pericytes; volume of immature and mature vessels;and concentrations of the regulatory factors such as VEGF, PDGF, Ang1and Ang2.

Moreover, several relative parameters (ratios) are calculated, such asAng2/Ang1, immature vessel density and mature vessel density (denotingvessels volume divided by tissue volume). The latter two densities arecombined into effective vessel density, EVD. EVD is a critical modelvariable, which at any moment determines tissue cells proliferation anddeath, as well as the production of factors, such as VEGF and PDGF.Resistance of tissue cells to antiangiogenic drugs may emerge fromtissue adaptation to hypoxia.

In order to account for the possible adaptation of tissue cells toinsufficient nutrition and to hypoxia it is assumed that EVD is afunction of the duration of insufficient perfusion, (denoted below byEVDn).

The technique takes into account the threshold levels of regulatoryfactors and parameter ratios, such as:

VEGF concentration below which no endothelial cells proliferation takesplace (denoted below by A);

The minimum number of receptors for VEGF above which endothelial cellsproliferation takes place(denoted below by B);

VEGF concentration below which endothelial cells (both in the free stateas well as when incorporated into immature blood vessels) are subject toapoptosis (this is denoted below by VEGF_(thr);

The minimal number of free pericytes which stimulates the onset ofmaturation of immature vessels(denoted below by C);

The Ang 1/Ang 2 ratio below which mature vessels are destabilized, andabove which maturation of immature vessels is enabled(denoted below byK).

The EVD value influences the rate of cell proliferation. The EVD valuefor which the system is in steady state (tissue cell proliferation ratebeing equal to tissue cell death rate) is denoted below by EVD_(ss). AtEVD>EVD_(ss) tissue cells proliferation prevails, so that tissue volumeincreases. At EVD<EVD_(ss) tissue cell death prevails, and the tissueshrinks. The EVD_(ss) is determined by genetic properties of a giventissue and a given host. VEGF, PDGF, Ang 1, Ang2 secretion level at thesteady state of the system will be denoted VEGF_(ss) PDGF_(ss),Ang1_(ss) and Ang2_(ss).

The inputs to the represented system includes the tissue volume, bloodvessels density, and the inherent parameters characterizing this tissuetype at initiation of the process. The output at any given moment areparameters like tissue volume, mature and immature vessels sizes, EVD.

IV.B. Detailed Description of the Exemplary Implementation

The flowchart shown in FIG. 1 is discussed in detail herein withreference to specific mathematical equations describing the principalinteractions affecting vascular tissue growth. The technique describesthe interrelationships between tissue growth, the formation of newvessels (angiogenesis) and the maturation of the newly formed vessels.The interactions occur across three organization levels: molecular,cellular, and organic level. The arrows in the flowchart indicate thespecific module interaction. The rectangular boxes indicate the point atwhich a specific sub-process calculation occurs. The parameter Tx in abox denotes the characteristic reaction time of the action calculated inthe box. The diamonds indicate the conditions, which determine thedirection of processes.

EVD_(ss) is the value for which the system is in steady state. VEGF_(ss)is the VEGF secretion level at the steady state of the system.VEGF_(Thr) is the VEGF concentration below which endothelial cells, bothin the free state as well as when incorporated into immature bloodvessels, are subject to apoptosis. PDGF_(ss) is the PDGF secretion levelat the steady state of the system.

In this mathematical model EVD_(n) in a certain moment n is representedas the sum of a density of immature (EVD_(n) ^(im)) and density ofmature vessels ( EVD_(n) ^(mat)) at the moment “n”.

EVD _(n) =EVD _(n) ^(im) +EVD _(n) ^(mat);  (1)

In FIGS. 1 and 2 the effective vessel density as discussed above iscalculated in block 1.1. The mature and immature vessel densities, inturn, are calculated in blocks 1.2 and 1.3 using the followingequations:

$\begin{matrix}{{{EVD}_{n}^{im} = \frac{\alpha^{im}*{Vves}_{n}^{im}}{{Vtis}_{n}}};{{EVD}_{n}^{mat} = \frac{\alpha^{mat}*{Vves}_{n}^{mat}}{{Vtis}_{n}}};} & (2)\end{matrix}$

The EVD_(n) ^(im) and EVD_(n) ^(mat) is the relation of volume ofvessels feeding the tissue, to a number of living tissue cells. Theamount of immature vessels at a moment “n” depends on an amount of bothimmature and mature vessels at the previous moment “n−1”.

All above described processes have an effect on the changes of theamount of vessels. The volume of immature vessels (block 1.4) at themoment “n” is a function of the volumes of immature and mature vesselsat the moment “n−1”. This function has 5 terms, corresponding to thefive exponential terms below. They are computed in blocks 1.5, 1.6, 1.7,1.8 and 1.9.

$\begin{matrix}{\left. {V_{n}^{im} = {{V_{n - 1}^{im}*^{{(\begin{matrix}{{A_{n - 1}^{{im}\Rightarrow{new}}\frac{T_{0}}{T_{3}}} -} \\{{A_{n - 1}^{{im}\Rightarrow{reg}}\frac{T_{0}}{T_{4}}} - {A_{n - 1}^{{im}\Rightarrow{mat}}\frac{T_{0}}{T_{6}}}}\end{matrix})}*\phi}} + {V_{n - 1}^{m}*^{{(\begin{matrix}{{A_{n - 1}^{{mat}\Rightarrow{new}}\frac{T_{0}}{T_{5}}} +} \\{A_{n - 1}^{{mat}\Rightarrow{im}}\frac{T_{0}}{T_{7}}}\end{matrix})}*\phi}} - 1}} \right);} & (3)\end{matrix}$

The generation of immature vessels by immature vessels (A^(im)

^(new)) is accounted for by block 1.5. The generation of immaturevessels by mature vessels (A^(mat)

^(new))is accounted for by block 1.6. The destabilization of maturevessels (A^(mat)

^(im)) is accounted for by block 1.7. The maturation of immature vessels(A^(im)

^(mat)) is accounted for by block 1.8. The degeneration of immaturevessels (A^(im)

^(reg)) by is accounted for by block 1.9. The volume of mature vessels(block 1.10) at the moment “n” is also a function of the volumes ofimmature and mature vessels at the moment “n−1”. This function has 2terms corresponding to the two exponential terms as shown below. Theyare calculated in blocks 1.7 and 1.9 respectively.

$\begin{matrix}{{V_{n}^{m} = {{V_{n - 1}^{im}*\left( {^{A_{n - 1}^{{im}\Rightarrow{mat}}\frac{T_{0}}{T_{6}}*\phi} - 1} \right)} + {V_{n - 1}^{m}\left( {2 - ^{A_{n - 1}^{{mat}\Rightarrow{im}}\frac{T_{0}}{T_{7}}*\phi}} \right)}}};} & (4)\end{matrix}$

The maturation of immature vessels (A^(im)

^(new)) is accounted for by block 1.7 and the destabilization of maturevessels (A^(mat)

^(im)) is accounted for by block 1.9.

Every sub process described in functions (3) and (4) has itscharacteristic time, denoted by T₁ to T₇. Resolution is denoted by T₀(the period between “n” and “n−1”). Factor φ represents a factor of theconformity.

The terms in eqns. (3) and (4) are functions of the followingconcentrations: the generation of immature vessels is a function of theconcentration of VEGF with the coefficient λ_(im) ^(ec), λ_(mat) ^(ec),μ_(ec) and ρ_(ec) ^(im) and Eqns. (5) and (6);

$\begin{matrix}\left\{ \begin{matrix}{A_{n - 1}^{{im}\Rightarrow{new}} = {\rho_{ec}^{im}*\left( {{\lambda_{im}^{ec}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \\{A_{n - 1}^{{im}\Rightarrow{new}} = {{0\mspace{14mu} {{IF}\left( {{\lambda_{im}^{ec}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \leq 0}} \\{A_{n - 1}^{{im}\Rightarrow{new}} = {{\rho_{ec}^{im}\mspace{14mu} {{IF}\left( {{\lambda_{im}^{ec}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \geq 1}}\end{matrix} \right. & (5)\end{matrix}$

The degeneration of immature vessels is also a function of theconcentration of VEGF, level VEGF_(thr), with the coefficient μ_(im).

$\begin{matrix}\left\{ \begin{matrix}{A_{n - 1}^{{mat}\Rightarrow{new}} = {\rho_{ec}^{im}*\left( {{\lambda_{mat}^{{ec}\,}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \\{A_{n - 1}^{{mat}\Rightarrow{new}} = {{0\mspace{14mu} {{IF}\left( {{\lambda_{mat}^{ec}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \leq 0}} \\{A_{n - 1}^{{mat}\Rightarrow{new}} = {{\rho_{ec}^{im}\mspace{14mu} {{IF}\left( {{\lambda_{mat}^{ec}*\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}} - {\mu_{ec}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}} \right)}} \geq 1}}\end{matrix} \right. & (6)\end{matrix}$

The destabilization of mature vessels (block 1.9) is a function of theratio between Ang1 and Ang2 with the coefficient μ_(mat) ^(im), Eqn.(8).

$\begin{matrix}{{A_{n - 1}^{{im}\Rightarrow{reg}} = {\mu_{im}/\frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}}};} & (7) \\{{A_{n - 1}^{{mat}\Rightarrow{im}} = {\mu_{mat}^{im}/\frac{{Ang}\; 1_{n - 1}}{{Ang}\; 2_{n - 1}}}};} & (8)\end{matrix}$

The maturation of immature vessels (block 1.7) is a more complicatedfunction, Eq. (9).

$\begin{matrix}{{A_{n - 1}^{{im}\Rightarrow{mat}} = {\lambda_{im}^{mat}*\frac{{Ang}\; 1_{n - 1}}{{Ang}\; 2_{n - 1}}*{\left( \frac{{Nper}_{n - 1}}{\rho_{mat}^{per}} \right)/V_{n - 1}^{im}}}};} & (9)\end{matrix}$

Maturation in a given moment is a function of a ratio of Ang1/Ang2 atthe same moment, with the coefficient λ_(im) ^(mat). Maturation is alsoa function of the volume of immature vessels and of the number of freepericytes. The term (N_(per)/ρ_(mat) ^(per))/V^(im) gives the fractionof immature vessels can potentially mature (If (N_(per)/ρ_(mat)^(per))/V^(im)>=1 then all immature vessels can mature).

The egn. 10 shows the functional dependence VEGF from EVD withcharacteristic time T₈ and T_(n−1) ^(VEGF).

$\begin{matrix}{{VEGF}_{n - 1} = {{V_{n - 1}^{tis}*{\lambda_{EVD}^{VEGF}\left( {{EVD}_{b} - {EVD}_{n - 2}} \right)}*\frac{T_{0}}{T_{8}}} + {{VEGF}_{n - 2}*^{(\begin{matrix}{{- 0.693171805}*} \\\frac{T_{0}}{T_{h - 1}^{VEGF}}\end{matrix})}}}} & (10)\end{matrix}$

IF VEGF_(n−1)>VEGF_(max) THEN VEGF_(n−1)=VEGF_(max)

IF VEGF_(n−1)<VEGF_(en) THEN VEGF_(n−1)=VEGF_(en)

The initial level VEGF_(ss) characterizes the amount of VEGF secretedwhen effective tissue vessel density is EVD_(ss).

In a similar way we obtain the dependence of PDGF (11).

$\begin{matrix}{{PDGF}_{n - 1} = {{V_{n - 1}^{tis}*{\lambda_{EVD}^{PDGF}\left( {{EVD}_{b} - {EVD}_{n - 2}} \right)}*\frac{T_{0}}{T_{14}}} + {{PDGF}_{n - 1}*^{(\begin{matrix}{{- 0.693171805}*} \\\frac{T_{0}}{T_{h - 1}^{PDEGF}}\end{matrix})}}}} & (11)\end{matrix}$

IF VEGF_(n−1)>VEGF_(max) THEN VEGF_(n−1)=VEGF_(max)

IF VEGF_(n−1)<VEGF_(en) THEN VEGF_(n−1)=VEGF_(en)

The characteristic time T₁₄ and T_(n−1) ^(PDGF). The above equations 10and 11 are involved in blocks marked 1.11 & 1.12.

$\begin{matrix}{{{Ang}\; 2_{n - 1}} = {{\begin{pmatrix}{{\left( {{{Ang}\; 2_{en}^{ec}} + {{Ang}\; 2_{ss}^{ec}}} \right)*{EC}_{n - 2}} +} \\{{Ang}\; 2_{ed}^{ec}*\left( {{EDV}_{ss} - {EVD}_{n - 2}} \right)*{EC}_{n - 2}}\end{pmatrix}*{\frac{T_{0}}{T_{10}}++}\begin{pmatrix}{{\left( {{{Ang}\; 2_{en}^{tc}} + {{Ang}\; 2_{ss}^{tc}}} \right)*N_{n - 2}^{tum}} +} \\{{Ang}\; 2_{ed}^{tc}*\left( {{EVD}_{b} - {EVD}_{n - 2}} \right)*N_{n - 2}^{tum}}\end{pmatrix}*\frac{T_{0}}{T_{15}}} + {{Ang}\; 2_{n - 2}*^{(\begin{matrix}{{- 0.693171805}*} \\\frac{T_{0}}{T_{h - 1}^{{Ang}\; 2}}\end{matrix})}}}} & (12)\end{matrix}$

In Eqn. (12, 14) Ang2 and Ang1 also depends on the numbers ofendothelial cells in immature vessels, which is determined by the Eqs.(13), and the numbers of tissue cells Ang2_(en) ^(ec), Ang2_(b) ^(ec),Ang2_(ed) ^(ec), Ang2_(en) ^(tc), Ang2_(b) ^(tc), Ang2_(ed) ^(tc),T_(n−1) ^(Ang2), ρ_(V) _(im) ^(ec), Ang1 _(en) ^(ec), Ang1_(b) ^(ec),Ang1_(ed) ^(ec), Ang1_(en) ^(tc), Ang1_(b) ^(tc), Ang1_(ed) ^(tc),T_(n−1) ^(Ang1). The Ang1 induction and Ang2 induction are factors inbocks 1.4, 1.14 and 1.15.

EC _(n−1) =

K ₅

ρ_(V) _(im) ^(ec) *V _(n−1) ^(im);

The characteristic reaction time for Ang2 generation is T₁₀ and T₁₅,(12), for Ang1 generation it is T₉ and T₁₁ (14).

$\begin{matrix}{{{Ang}\; 1_{n - 1}} = {{\begin{pmatrix}{{\left( {{{Ang}\; 1_{en}^{tc}} + {{Ang}\; 1_{ss}^{tc}}} \right)*N_{n - 2}^{tum}} +} \\{{Ang}\; 1_{ed}^{tc}*\left( {{EVD}_{ss} - {EVD}_{n - 2}} \right)*N_{n - 2}^{tum}}\end{pmatrix}*{\frac{T_{0}}{T_{9}}++}\begin{pmatrix}{{\left( {{{Ang}\; 1_{en}^{ec}} + {{Ang}\; 1_{ss}^{ec}}} \right)*{EC}_{n - 2}} +} \\{{Ang}\; 1_{ed}^{ec}*\left( {{EVD}_{ss} - {EVD}_{n - 2}} \right)*{EC}_{n - 2}}\end{pmatrix}*\frac{T_{0}}{T_{16}}} + {{Ang}\; 1_{n - 2}*^{(\begin{matrix}{{- 0.693171805}*} \\\frac{T_{0}}{T_{h - 1}^{{Ang}\; 1}}\end{matrix})}}}} & (14)\end{matrix}$

The addition of free pericytes (block 1.17) at any given moment dependson the level of PDGF at the previous moment, the replication of freepericytes, and on the number of free pericytes released from maturevessels (15). Accordingly, these two processes have the coefficientsλ_(bou) ^(per) and λ_(fr) ^(per). It is also necessary to take intoaccount the characteristic reaction time of these processes T₁₂ and T₁₃.

$\begin{matrix}{{Nper}_{n - 1} = {{PDGF}_{n - 2}*\left( {{\lambda_{bou}^{per}*V_{n - 2}^{mat}*\frac{T_{0}}{T_{12}}} + \left( {{Nper}_{n - 2} - {\rho_{mat}^{per}*\left( {V_{n - 1}^{mat} - V_{n - 2}^{mat}} \right)*\lambda_{fr}^{per}*\frac{T_{0}}{T_{13}}}} \right)} \right.}} & (15)\end{matrix}$

The number of tissue cells (block 1.18) in the moment, n, depends ontheir number in the previous moment multiplied by a factor describingthe process of cell proliferation and death, r_(n−1).

V _(n) ^(tis) =V _(n−1) ^(tis) *e ^(r) ^(n−1) ^(*φ)  (16)

r_(n−1) depends on the mitotic index M(I) (mitotic time being T(1),apoptotic index A (I) (apoptotic time being T (2)), rate of tissue cellgrowth λ and the rate of the death of tumor cells, μ. The two terms inthe equation below are involved in blocks 1.191 and 1.201 respectively.Clearly, they are also factors in blocks 1.192 and 1.202 as well as1.193 and 1.203.

$r_{n - 1} = {{\left( {M_{I} + \lambda_{n - 1} - ɛ_{1}} \right)*\frac{T_{0}}{T_{1}}} - {\left( {A_{I} + \mu_{n - 1} + ɛ_{2}} \right)*\frac{T_{0}}{T_{2}}}}$

The proliferation rate λ and the death rate μ are assumed to be standardsigmoids. Hence we obtain λ and μ as follows:

$\begin{matrix}{\mu_{n - 1} = {1 - A_{I} - \frac{\left( {1 + ɛ_{2}} \right)*{EVD}_{n - 1}^{\frac{1 + ɛ_{2}}{{2*A_{I}} - 1 + ɛ_{2}}}}{\begin{matrix}{{EVD}_{n - 1}^{\frac{1 + ɛ_{2}}{{2*A_{I}} - 1 + ɛ_{1}}} +} \\\left( {\frac{A_{I} + ɛ_{2}}{1 - A_{I}}*{EVD}_{ss}^{\frac{1 + ɛ_{2}}{{2*A_{I}} - 1 + ɛ_{1}}}} \right)\end{matrix}}}} & (18) \\{\lambda_{n - 1} = {\frac{\left( {1 + ɛ_{1}} \right)*{EVD}_{n - 1}^{\frac{1 + ɛ_{1}}{1 - {2*M_{I}} + ɛ_{1}}}}{\begin{matrix}{{EVD}_{n - 1}^{\frac{1 + ɛ_{1}}{1 - {2*M_{I}} + ɛ_{1}}} +} \\\left( {\frac{1 - M_{I} + ɛ_{1}}{M_{I}}*{EVD}_{ss}^{\frac{1 + ɛ_{1}}{1 - {2*M_{I}} + ɛ_{1}}}} \right)\end{matrix}} - M_{I}}} & (19)\end{matrix}$

IV.C. Tissue Control by Pro and Anti Angiogenic Drugs.

Possible drug effects on the pro and anti angiogenesis process indicatedin FIG. 2. Note that the blocks in FIG. 2 are identical to those in FIG.1, except for the additional drug effects shown. The drug effects on theoverall process are analyzed by setting the selected drug schedule(number of doses, the dose and the dosing interval). For example theanalysis of anti-VEGF drug activity shows that a drug which inhibitsVEGF has an optimal efficacy when given by certain treatment protocol.Increasing the administered dose above the optimum can bring about theundesired effect of tissue proliferation as shown in FIG. 3. Inaddition, the technique enables one to predict the effects of variousdrug combination, for example as shown in FIG. 4.

Other modifications and variations to the invention will be apparent tothose skilled in the art from the foregoing disclosure and teachings.Thus, while only certain embodiments of the invention have beenspecifically described herein, it will be apparent that numerousmodifications may be made thereto without departing from the spirit andscope of the invention.

1. A computer-implemented method for determining an optimal treatmentprotocol for a disease related to angiogenesis, comprising: creating anangiogenesis model including pro-angiogenesis and anti-angiogenesisfactors; incorporating effects of vasculature maturation and maturevessels destabilization; selecting pro-angiogenesis andanti-angiogenesis factors, which can influence changes in state of atissue; incorporating effects of drugs in the pro-angiogenesis andanti-angiogenesis factors; generating a plurality of treatment protocolsin a protocol space; and selecting a best treatment protocol based on apre-determined criteria.
 2. A system for determining an optimaltreatment protocol for a disease related to angiogenesis, comprising: anangiogenesis model including pro-angiogenesis and anti-angiogenesisfactors; a treatment protocol space generator that generates a protocolspace of possible treatments for the disease; a treatment selector thatselects an optimal protocol, wherein the model incorporates effects ofvasculature maturation and mature vessels destabilization; wherein thesystem is adapted to effect selection of pro-angiogenesis andanti-angiogenesis factors which can influence changes in state of atissue and incorporating effects of drugs in the pro-angiogenesis andanti-angiogenesis factors.
 3. A computer program product, includingcomputer-readable media comprising instructions to implement proceduresfor determining an optimal treatment protocol for a disease related toangiogenesis, said procedure comprising: creating an angiogenesis modelincluding pro-angiogenesis and anti-angiogenesis factors; incorporatingeffects of vasculature maturation and mature vessels destabilization;selecting pro-angiogenesis and anti-angiogenesis factors, which caninfluence changes in state of a tissue; incorporating effects of drugsin the pro-angiogenesis and anti-angiogenesis factors; generating aplurality of treatment protocols space; and selecting a best treatmentprotocol based on a pre-determined criteria.